Appl. Sci. 2015, 5, 157-173; doi:10.3390/app5030157
applied sciences ISSN 2076-3417
www.mdpi.com/journal/applsci
Article
Prediction of Experimental Rainfall-Eroded Soil Area Based on
S-Shaped Growth Curve Model Framework
Wen Nie 1,2,*, Run-Qiu Huang 2,†, Qian-Gui Zhang 3, Wei Xian 4, Feng-Lin Xu 3,† and
Lin Chen 5,†
1 State Key Laboratory of Coal Mine Disaster Dynamics and Control, Chongqing University,
Chongqing 400044, China 2 State Key Laboratory of Geo-hazard Prevention and Geo-environment Protection,
Chengdu University of Technology, Chengdu 610059, Sichuan, China; E-Mail: [email protected] 3 State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation,
Southwest Petroleum University, Chengdu, 610500, Sichuan, China;
E-Mails: [email protected] (Q.-G.Z.); [email protected] (F.-L.X.) 4 School of Aeronautics & Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China;
E-Mail: [email protected] 5 School of Science, Southwest Petroleum University, Chengdu 610500, Sichuan, China;
E-Mail: [email protected]
† These authors contributed equally to this work.
* Author to whom correspondence should be addressed; E-Mail: [email protected];
Tel.: +49-017631199092.
Academic Editor: Takayoshi Kobayashi
Received: 10 May 2015 / Accepted: 7 July 2015 / Published: 14 July 2015
Abstract: Rainfall-induced soil erosion of a mountain area plays a significant role in
supplying sediment and shaping the landscape. The related area of soil erosion, as an index
of the changed landscape, is easier to calculate visually using some popular imaging tools.
By image analysis, our work shows that the changing of the soil erosion area admits the
structure of an S-growth curve. Therefore, we propose to establish an S-curve model, based
on incremental learning, to predict the soil erosion area. In the process of incremental
learning, we dynamically update the accumulative rainfall and rainfall intensity to train the
parameters of our S-curve model. In order to verify our prediction model, the index of area
is utilized to express the output of eroded soil in a series of experiments. The results show
that the proposed S-growth curve model can be used to estimate the growth of the soil
OPEN ACCESS
Appl. Sci. 2015, 5 158
erosion area (average relative error 3%–9.7%) according to variable soil material and
rainfall intensity. The original S-growth curve model can calculate the erosion areas of just
one soil material and one rainfall condition whose average relative error is 7.5%–12.2%;
compared to the simple time series analysis-moving average method (average relative error
5.7%–12.1%), our proposed S-growth curve model can reveal the physical mechanism and
evolution of the research object.
Keywords: eroded soil area; S-shaped growth curve; time series analysis;
incremental learning
1. Introduction
Rainfall-induced soil erosion involves three main processes: detachment, transport, and deposition
of soil particles by the crush forces of rainfall and surface runoff. During the erosion process, the area
of soil erosion can reflect the shaped landscape. The classical estimation models of soil erosion include
the USLE model, which is composed of six factors to predict the long-term average annual soil
weight [1,2]; the WEPP model, which is capable of predicting spatial and temporal distributions of soil
detachment and deposition on an event or continuous basis at both small (hillslopes, roads, small
parcels) and large (watershed) scales [3]; the EUROSEM model, which is a dynamic distributed
model, able to simulate sediment transport, erosion and deposition by rill and inter-rill processes in
single storms for both individual fields and small catchments [4]; the KINEROS model, which is an
event-oriented, physically-based model describing the processes of interception, infiltration, surface
runoff and erosion from agricultural and urban watersheds [5]. These models can commonly describe
the distributions of soil detachment in large (watershed) scales. By contrast, in our study, a narrow
scale of even a single slope is our research object. We aim to investigate the development of the area
of soil erosion induced by rainfall in a single slope using an imaging tool. In the practical experiments,
the values of rainfall intensity and cumulative rainfall are acquired incrementally with respect to time.
The area of soil erosion is achieved by the imaging tool under time domains. Then, in this work, based
on incremental learning, we develop an S-curve model to predict the soil erosion area. In detail, we
update the input values chronologically to train the parameters of the S-curve model. Five experiments
involving different materials and different rainfall intensities are employed to validate the model.
2. Physical Experiments and Results
2.1. Physical Models, Monitoring Devices, and Experimental Procedures
Schmidt et al. [6] highlight the advantage of physically-based models: using minimal historical data
to produce meaningful results. Thus, a trend of developing and testing physically-based soil erosion
models appears [7–11]. In our study, a slope model is constructed in a flume device made of Plexiglas
(Figure 1). The slope inclination is 34° and the material is from Fengdu Mountain, near to Yangtze
River Bank, Chongqing, China, whose surface soil can be easily eroded under heavy rainfall. The soil
was air-dried and crushed to different test samples having three particle size distributions, as shown in
Appl. Sci. 2015, 5 159
Figure 2. The soil sample was squeezed by a plate with a 120 N force to form the slope.
In order to reduce the “boundary effect”, we used a covering of polytetrafluoroethylene (PTFE) whose
friction coefficient is 0.04 on both sides of the flume. In Figure 1, m is the moisture transducer
(frequency domain sensor) [12] whose contact area with the soil is less than 20 mm2 (measuring range
0%–100%, resolution 0.1%, deviation ±2%); p is the pore water pressure transducer (diameter
3 cm, height 1.6 cm, measuring range ±10 kPa, deviation ±0.2%); d is a channel for drainage;
one high-definition digital video camera (5 million pixels, position in Figure 1) is used to record the
changes of the eroded soil area (front scene). Every test is conducted under similar initial geometry,
and pre-added water content (deviation ±3%, average value ~5.2%). Persistent rainfall is averagely
simulated by three nozzle spray heads (~0.6 m high to slope surface) in five rainfall events. For soil
Sample 1, we used three rainfall intensity levels in three experiments (approximately 25, 45, and
65 mm/h, respectively) to investigate the effect of different rainfall intensities on the same material.
Then we focused on the effect of the different materials (Sample 1, Sample 2, and Sample 3), by
comparing their responses in experiments with a common rainfall intensity of 45 mm/h.
70cm
Camera
70cm
Camera
Figure 1. Geometry of physical model.
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0.01 0.1 1 10Particle size(mm)
Perc
enta
ge p
assi
ng(%
)
Soil sample1Soil sample2Soil sample3
Figure 2. Particle-size distribution curves.
Appl. Sci. 2015, 5 160
2.2. Soil Erosion Area Calculation
A fast way to calculate a two-dimensional area of an object is to use the “BOUNDARY” command
in the AutoCAD software (Autodesk, Inc., San Rafael, CA, United States of America) [13]. Using
“BOUNDARY”, a point can be picked within the area to create a closed poly line or polygon domain.
The “Features” palette can be used to find the area and perimeter of the poly line or polygon domain.
By using the “MEASUREGEOM” and “AREA” commands, a series of points or a selected object is
specified to calculate the area. In Figure 3a, a coordinate is first established. Then the contour of the
eroded soil area is extracted in Figure 3b. Finally, in Figure 3c, calculation of the selected area is
executed by AutoCAD. In every experiment, the video data per minute is chronologically intercepted
to calculate the area. Lens distortions and the distance between camera and ground affect the accuracy
of the picture area. An error calculation is carried out in Figure 3a, and it is found that the blue part,
which concentrates on the upper part is a picture area error of ~4.2%, while most of the soil erosion
happens in the lower part, which means that the area error of soil erosion could be less than 4.2%.
y=130cm
x=55cm
error~4.2%
y=130cm
x=55cm
y=130cm
x=55cm
error~4.2%
(a) (b) (c)
Figure 3. Area calculation by AutoCAD (soil Sample 1 and rainfall intensity-65 mm/h):
(a) establishment of coordinate; (b) extraction of contour; and (c) calculation of areas.
2.3. Soil Erosion Area
Increased detachability produced by surface runoff shear forces and the impact of raindrops is
considered as the main force driving erosion [14]. Figure 4 illustrates that the toe of the slope is more
eroded than other positions, in consideration of more ground water supply from the upper parts. The
erosion of the top part of the slope without ground water supply from other parts shows a slow increase.
Thus, this is typical progressive soil erosion. Figure 5 shows the erosion of different materials under
the same time domain and rainfall intensity. It is found that the coarse particles (such as soil Sample 3)
have lower erosion than fine particles (such as soil Sample 1) because of their higher permeability and
stronger framework.
Appl. Sci. 2015, 5 161
(a) (b) (c) (d)
Figure 4. Different time domains in Sample 1 and rainfall intensity-65 mm/h test:
(a) t = 6 min; (b) t = 19 min; (c) t = 45 min; and (d) t = 70 min.
(a) (b) (c)
Figure 5. Different materials under the same time domain (45 mm/h; t = 52 min):
(a) soil Sample 1 and erosion area-771.21 mm2; (b) soil Sample 2 and erosion
area-421.31 mm2; (c) soil Sample 3 and erosion area-359.46 mm2.
In Figure 6, soil Sample 1 (25 mm/h) has the most obvious initial phase (threshold of soil erosion
action) because of the low rainfall intensity. By comparison, soil Sample 1 (45 mm/h) and soil
Sample 1 (65 mm/h) have fewer time lags. In the second acceleration stage, the erosion under higher
rainfall intensity obviously occurs faster because of the greater surface runoff production. In the last
phase, for one thing the coarse particles remain and the loss of fine particles makes the soil structure
stronger; for another, the water erosion develops towards the deep direction. Therefore, the soil erosion
area has a maximum value, but the volume could still increase. By contrast, for different materials,
soil Sample 1 (more fine particles), with the weakest structure, has the earliest start point of the
second erosion phase and a high erosion rate during the second stage because more surface fine
particles are lost. Until the final stage (relatively stable structure), the erosion rate is still higher than
for the coarse materials, but only towards the deep direction. For the area of soil erosion in the final
stage, the values of the three materials can be close, while for the erosion volume or weight the coarse
material has a lower value. This is because coarse particles like soil Sample 3 have a stronger structure
(high friction) and higher infiltration rate (producing less surface runoff) than fine particles like soil
Appl. Sci. 2015, 5 162
Sample 1. In addition, it can also be seen that the stronger the structure of coarse material against soil
erosion is, the greater is the time lag of the beginning and accomplishment of the erosion area compared to
fine material. In summary, the soil erosion is decided by both rainfall and the material’s strength.
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0 20 40 60 80 100 120 140 160 180 200
Time(min)
Are
a(m
m2)
soil sample1-25mm/hr soil sample1-45mm/hr soil sample1-65mm/hr
soil sample2-45mm/hr soil sample3-45mm/hr
Figure 6. Results of soil erosion experiments.
2.4. Hydrological Characteristics
Subsurface water content and soil pore water pressure might reflect the effects of rainfall on the
processes of surface sealing, runoff generation and soil sediment production. For one thing, high soil
moisture content or low negative pore water pressure results in an increased detachability by surface
runoff shear forces and raindrop impact [14]. For another aspect, a high water content of the soil might
decrease aggregate slaking and breakdown due to less air escaping upward, reducing the detachment of
soil by raindrop impact [15,16]. In Figure 1, m1 is at the middle deep position; m2 is near to the
surface; and m3 is at the low part. By contrast, p1 is at the middle depth position; p2 is near to the
surface; and p3 is at the toe of the slope. Figure 7a demonstrates that m2 near to the surface increases
more easily. Due to their depth, m1 and m3 have a certain time lag before increasing of water content.
Furthermore, m3 has a higher value and a shorter time lag because of the water flow supply.
In Figure 7b, the pore water pressure follows a similar pattern. Point p2, at the surface, is always
saturated to some degree and has no potential to increase. Point p3 has a higher value than p1 because
of its deeper position. For the same material, a higher rainfall intensity means more water input.
Therefore, m and p have higher values and lower time lags in Figures 8 and 9. For the different
materials, in Figures 8, 10 and 11 the coarse particles in soil Sample 3 have higher permeability or a
higher infiltration rate than fine particles. Thus, the time lag of pore water pressure production in
coarse materials is shorter and the value is higher. The moisture value depends on the balance of the
water infiltration, supply, and drainage.
Appl. Sci. 2015, 5 163
0
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1 13 25 37 49 61 73 85 97 109 121 133 145 157 169
Time(min)
Mo
istu
re(%
)
m1m2m3
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Time(min)
Po
re w
ate
r p
ress
ure
(kP
a)
p1p2p3
(a) (b)
Figure 7. Soil Sample 1-25 mm/h: (a) Moisture content of m1, m2, and m3; and
(b) pore water pressure of p1, p2, and p3.
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0 25 50 75 100 125 150 175 200
Time(min)
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re w
ate
r p
ress
ure
(kP
a)
p1p2p3
(a) (b)
Figure 8. Soil Sample 1-45 mm/h: (a) Moisture content of m1, m2, and m3; and
(b) pore water pressure of p1, p2, and p3.
0
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1 13 25 37 49 61 73 85 97
Time(min)
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istu
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m1m2m3
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1 13 25 37 49 61 73 85 97
Time(min)
Po
re w
ate
r p
ress
ure
(kP
a)
p1p2p3
(a) (b)
Figure 9. Soil Sample 1-65 mm/h: (a) Moisture content of m1, m2, and m3; and
(b) pore water pressure of p1, p2, and p3.
Appl. Sci. 2015, 5 164
0
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1 13 25 37 49 61 73 85 97 109 121 133 145 157
Time(min)
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Time(min)
Po
re w
ate
r p
ress
ure
(kP
a)
p1p2p3
(a) (b)
Figure 10. Soil Sample 2-45 mm/h: (a) Moisture content of m1, m2, and m3; and
(b) pore water pressure of p1, p2, and p3.
0
2
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6
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1 13 25 37 49 61 73 85 97 109 121 133 145 157 169 181 193
Time(min)
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istu
re(%
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m1m2m3
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1 13 25 37 49 61 73 85 97 109 121 133 145 157 169 181 193
Time(min)
Po
re w
ate
r p
ress
ure
(kP
a)
p1p2p3
(a) (b)
Figure 11. Soil Sample 3-45 mm/h: (a) Moisture content of m1, m2, and m3; and
(b) pore water pressure of p1, p2, and p3.
3. Soil Erosion Area Prediction Model
3.1. S-Shaped Growth Curve
The S-curve is one of the most common phenomena in nature: It is seen in the spreading of
populations, tumors, contaminants, innovations, and economic activity [17–20]. The pattern of an
S-shaped growth curve (Pearl growth curve) usually comprises three phases in an environment: the
object initially increases slowly; then it enters a positive acceleration phase, increasing rapidly and
approaching an exponential growth rate as in the J-shaped curve; and finally, the object declines in a
negative acceleration phase until reaching almost zero growth rate (logistic function).
y =
k
1+ be-ax,a > 0,k > 0,x > 0 (1)
As shown in Equation (1), this slowing of the growth rate reflects increasing environmental
resistance, which becomes proportionately more important at higher object densities. The point of
Appl. Sci. 2015, 5 165
stabilization, or zero growth rate, is termed the “saturation value” (symbolized by k) or “carrying
capacity” of the environment. a and b are the parameters that decide the trend of curve development;
y is the output object; and x is the input variable. It is usually summarized mathematically by the
logistic equation. In our case, when the cumulative rainfall (x) achieves a certain threshold, erosion
begins to occur. Thus, in the first stage, the eroded area (y) increases slowly, even having some time
lag, especially under low rainfall intensity; in the second phase, the eroded area increases swiftly due
to the shear force produced by surface runoff; in the final phase, because of surface loss of weak
particles while strong particles remain, the erosion develops deeper into the slope (“carrying capacity”
of the environment (k)), and the eroded area growth rate then begins to reduce, even to zero.
3.2. S-Shaped Growth Curve Fitting
According to Equation (1), the S-curves of the five experiments are fitted. In Equation (1), x is
cumulative rainfall; y is the eroded area; k is the maximum eroded area; a is a parameter related with
material; and b is a parameter to control the erosion rate. Figure 12 shows that the S-curve model
matches the soil erosion area development with a relative average error of 7.5%–12.2%. Table 1
presents the parameters in the five models.
0
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0 500 1000 1500 2000 2500 3000 3500 4000 4500Cumulative rainfall (mm)
Are
a(m
m2)
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Observation
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a(m
m2)
Fitting
Observation
(a) (b)
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m2)
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Observation
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0 500 1000 1500 2000 2500 3000 3500 4000 4500Cumulative rainfall (mm)
Are
a(m
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Fitting
Observation
(c) (d)
Figure 12. Cont.
Appl. Sci. 2015, 5 166
0
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0 800 1600 2400 3200 4000 4800 5600 6400Cumulative rainfall(mm)
Are
a(m
m2)
Fitting
Observation
(e)
Figure 12. S-growth curve fitting: (a) soil Sample 1-25 mm/h; (b) soil
Sample 1-45 mm/h; (c) soil Sample 1-65 mm/h; (d) soil Sample 2-45 mm/h; and
(e) soil Sample 3-45 mm/h.
Table 1. Parameters of five S-shaped growth curves.
Name k a b
Soil Sample 1-25 mm/h 3000 1.088 × e4 10.4
Soil Sample 1-45 mm/h 6350 4.75 × e5 15.2
Soil Sample 1-65 mm/h 9250 4.215 × e5 1.176
Soil Sample 2-45 mm/h 6350 4.4 × e5 20.6
Soil Sample 3-45 mm/h 6350 4.267 × e5 16.9
3.3. Development of Prediction Model
In order to predict the S-shaped growth curve framework based on different soil materials under
different rainfall intensities, we used the previous observation values to train the parameters of a model
to predict the object in the next time domain and then repeat the process in the following time domain,
which is similar to the dynamically updated method. The modified S-shaped growth curve framework
is used to predict the eroded area 1ts at time t+1, as in Equation (2):
11 ( )
', 0, ' 0, 0
1
t t tt t ta x x
t
ks a k b
b e (2)
where ta , related to material, is the parameter affecting the erosion rate at time t. It is considered a
variable as time elapses because the particles of the soil change. tb is considered as the variable which
depends on the cumulative eroded area and rainfall intensity at time t. tx and 1tx are the cumulative
rainfall at time t and t−1, respectively.
In Equation (1), the initial value of 0b can be calculated when t is zero. Therefore,
b0= (
k '
s0
-1) (3)
where k’ is the “saturation value” or “carrying capacity” of the eroded area. From the results in Table
1, it is found that k is mainly affected by rainfall intensity m linearly, rather than the material parameter
a, which is read as
Appl. Sci. 2015, 5 167
' , 3.44 k λm μ m (4)
where m is the rainfall intensity and the coefficients λ (147.5) and μ (507.5) of a linear relationship
can be obtained by experimental data fitting. The minimum value of m is 3.44, meaning the threshold
of erosion beginning (in order to make k′ positive). If the detachment capacity is significantly lower
than the transport capacity, the process is referred to as “detachment-limited erosion” [21]. Materials
are transported only after being detached. Thus, it should have a threshold of the rainfall amount,
which is the start point of soil erosion. This agrees with the fact that the shear force induced by
runoff over the resistance force is a precondition of soil loss, as reported by The Water Erosion
Prediction Project [22] and SHETRAN [23]. The final model is as follows after putting Equations (3)
and (4) into Equation (5):
1
1( )
, 0, ' 0, 0
1 ( 1)
t t t
t t ta x x
t
λm μs a k s
λm μe
s
(5)
For the proposed model framework, we just need to input the rainfall intensity and the calculated
cumulative rainfall at time t without other parameters in order to obtain the eroded soil area at time
t+1. When new data at time t+1 is inputted, this model can predict the eroded soil area at time t+2.
Figure 13 indicates the five predictions of the S-growth curve model based on the physical
experiments. The prediction results match the observation values well and the average errors are
between 3% and 9.7%. The standard deviations Ns of the five time predictions are 1180, 1170, 1172,
1181, and 1174, respectively, according to Equation (6):
2
1
1( )
N
N ii
s x xN
(6)
Where ix is the observed value of the sample item and x is the mean value of these observations,
while the denominator N stands for the size of the sample: this is the square root of the sample variance,
which is the average of the squared deviations about the sample mean.
0 1000 2000 3000 4000 50000
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Cumulative rainfall(mm)
Are
a(m
m2)
Prediction
Observation
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6000
7000
Cumulative rainfall(mm)
Are
a(m
m2)
Prediction
Observation
(a) (b)
Figure 13. Cont.
Appl. Sci. 2015, 5 168
0 2000 4000 6000 8000 100000
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Cumulative rainfall(mm)
Are
a(m
m2)
Prediction
Observation
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Cumulative rainfall(mm)
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a(m
m2)
Prediction
Observation
(c) (d)
0 1000 2000 3000 4000 50000
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3000
4000
5000
6000
Cumulative rainfall(mm)
Are
a(m
m2)
Prediction
Observation
(e)
Figure 13. Prediction of model: (a) soil Sample 1-25 mm/h (n = 174); (b) soil
Sample 1-45 mm/h (n = 171); (c) soil Sample 1-65 mm/h (n = 142); (d) soil
Sample 2-45 mm/h (n = 107); and (e) soil Sample 3-45 mm/h (n = 177).
4. Discussion
4.1. Dimensionless Quantities of Soil Erosion Area
The developments of the soil erosion area are made as dimensionless plots as shown in Figure 14.
For different rainfall intensities, heavy rainfall still has a higher erosion rate; in the second phase, the
erosion rate from the light rainfall is greater than that induced by heavy rainfall; in the final stage, all
the soil erosion rates have similar trends. In other words, the soil erosion has a first low and then high
sensitivity to low rainfall intensity; the soil erosion has a first high and then low sensitivity to high
rainfall intensity; in the last stage, they have a similar sensitivity. For different materials, the erosion
rate of weak material (Sample 1) is higher than strong material in the first stage; in the second and
third phases, there is almost no difference between materials. Thus, it can be considered that the
sensitivity of the soil erosion area to material type is not obvious.
Appl. Sci. 2015, 5 169
0
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Cumulative rainfall
Are
a
soil sample1-25mm/hr soil sample1-45mm/hr
soil sample1-65mm/hr
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Cumulative rainfall
Are
a
soil sample1-25mm/hrsoil sample1-45mm/hrsoil sample1-65mm/hr
(a) (b)
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Cumulative rainfall
Are
a
soil sample1-45mm/hr soil sample2-45mm/hr
soil sample3-45mm/hr
0
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Cumulative rainfall
Are
a
soil sample1-45mm/hrsoil sample2-45mm/hrsoil sample3-45mm/hr
(c) (d)
Figure 14. Dimensionless plots of soil erosion area: (a) Soil erosion under different
rainfall intensity conditions; (b) soil erosion under different rainfall intensity conditions
(fitting curve); (c) soil erosion under different material conditions; and (d) soil erosion
under different material conditions (fitting curve).
4.2. Comparison Between S-Curve Based Model and Time Series Analysis–Moving Average Model
A case is investigated using a simple time series analysis–moving average method to predict the
eroded soil area in Equation (7) [24]. This method in Equation (7) uses the average of the previous
three values as the prediction value.
1 2 3' , 43
t t t
t
s s ss t (7)
where 'ts is the output object of the prediction model at time t. 1ts , 2ts , and 3ts are the observation
data at time t−1, t−2, and t−3, respectively. The result is as shown in Figure 15.
Appl. Sci. 2015, 5 170
0 1000 2000 3000 40000
500
1000
1500
2000
2500
3000
Cumulative rainfall(mm)A
rea
(mm
2)
Prediction
Observation
Figure 15. Prediction of moving averages method (Sample 1-25 mm/h) (n = 174).
Firstly, we did relative error analyses for the S-curve prediction model and moving averages model,
as shown in Figure 16 (for the case of sample1 under 25 mm/h rainfall). The average relative error of
all the experiments is 3%–9.7% for the S-curve prediction model. For the moving averages model, the
experimental average relative error is 5.7%–12.1%. The S-curve prediction model includes
information on the S-curve trend, and can thus describe the whole process, especially the changes of
phase. On the other hand, the moving averages model, whose prediction depends on the previous data,
cannot simulate the sudden change of Phase 1 to Phase 2. Thus, in the second stage, error exists all the
time until the final stage, reducing the increase of area. It has to be pointed out that the prediction
accuracy of the model still depends on the number of predictions. The greater the number of
predictions from the model, the lower its accuracy is, while the cost of accurate prediction is a smaller
number of forecasts. Secondly, the moving averages model is like a “black box”. Its use only involves
giving the input and getting the output without any clear physical meaning and process. In our study,
the S-growth curve model framework needs the rainfall intensity and cumulative rainfall (which can be
obtained from the rainfall intensity and time) as parameters. It can also describe the three phases of
eroded soil area growth from the physical angle. In addition, the model considers different soil
materials affecting the erosion area.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
25 325 625 925 1225 1525 1825 2125 2425 2725 3025 3325 3625 3925 4225
Cumulative rainfall(mm)
Re
lative
err
or
S-curve prediction model
Moving averages
Figure 16. Relative error analyses of S-curve prediction model and moving averages
model (Sample 1-25 mm/h).
Appl. Sci. 2015, 5 171
4.3. Limitation and Suggestion of Experiments
In the current stage, we did not consider the vegetation cover, slope effect and coupled with the
rainfall, material effect, because experiments are costly and very time-consuming. Slope shape and
surface roughness are relatively simple. As concerns the dimension effect, this is very common for any
mini-physical model experiment. In our case, the dimension effect arises mainly from the
underestimated rainfall impact and the “boundary effect”. The rainfall bump is reduced by the use of a
nozzle spray head in our experiment (we have revised this in Section 2.1). In order to reduce the
“boundary effect”, we use a covering of the polytetrafluoroethylene (PTFE), whose friction coefficient
is 0.04, on both sides of the flume. In the future, the impact on the soil erosion of the slope effect and
coupled the rainfall, slope, and material effect could be our next research direction.
5. Conclusions
In our work, we constructed a physical slope model to investigate to what extent different rainfall
intensities and slope materials affect the area of soil erosion acquired by an imaging tool. We further
presented an S-curve model to predict the growth of the eroded soil area, which is well trained
incrementally and verified efficiently by five physical experiments. The valuable conclusions include:
(1) The area of soil erosion obeys a trend of an S-growth curve under continuous rainfall.
(2) A higher rainfall intensity can produce a greater area of soil erosion than lower rainfall intensity
for the same material.
(3) Under the same rainfall intensity, coarse material (stronger material) has a time lag against the
soil erosion, but finally a similar area of soil erosion compared to fine material.
(4) The S-curve model coupled with incremental parameter learning can effectively predict the soil
erosion with just the rainfall information as input.
Acknowledgments
This research is supported by the Opening Fund of the State Key Laboratory of Geo-Hazard
Prevention and Geo-Environment Protection (Chengdu University of Technology)-SKLGP2013K007.
We are also grateful for support from the National Natural Science Foundation of China (51304170),
Young Scholars Development Fund of SWPU (201231010031).
Author Contributions
Drafting of manuscript: Wen Nie and Xian Wei; acquisition of data: Feng-Lin Xu; analysis and
interpretation of data: Qian-Gui Zhang and Wen Nie; Model construction: Xian Wei, Lin Chen,
Feng-Lin Xu and Wen Nie; and planning and supervision of the research: Run-Qiu Huang.
Appl. Sci. 2015, 5 172
Conflicts of Interest
The authors declare no conflict of interest.
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